Let $(X,x_0)$ be a pointed simplicial set. Assume if you like that $X$ is the nerve of a category but do not assume that $X$ is a Kan complex.
Because $Ex^\infty X$ is a Kan complex, every homotopy class $\alpha \in \pi_n(X,x_0)$ may be represented by a map $sd^k \Delta[n] \to X$ such that the restriction $sd^k \partial \Delta[n] \to X$ is constant at $x_0$. I'm wondering about different "normal forms" for homotopy classes.
For instance, consider subidivided cubes. In dimension 2, I think they should look like this:
$\require{AMScd} D^2_0 = \begin{CD} \bullet \end{CD} \\ D^2_1 = \begin{CD} \bullet @>>> \bullet @<<< \bullet \\ @VVV @VVV @VVV \\ \bullet @>>> \bullet @<<< \bullet \\ @AAA @AAA @AAA \\ \bullet @>>> \bullet @<<< \bullet \\ \end{CD} \\ D^2_2 = \begin{CD} \bullet @>>> \bullet @<<< \bullet @>>> \bullet @<<< \bullet\\ @VVV @VVV @VVV @VVV @VVV \\ \bullet @>>> \bullet @<<< \bullet @>>> \bullet @<<< \bullet\\ @AAA @AAA @AAA @AAA @AAA \\ \bullet @>>> \bullet @<<< \bullet @>>> \bullet @<<< \bullet\\ @VVV @VVV @VVV @VVV @VVV \\ \bullet @>>> \bullet @<<< \bullet @>>> \bullet @<<< \bullet\\ @AAA @AAA @AAA @AAA @AAA \\ \bullet @>>> \bullet @<<< \bullet @>>> \bullet @<<< \bullet \end{CD} \\ D^2_3 = \dots $
Questions:
- Can every $\alpha \in \pi_n(X, x_0)$ be represented by a map $D^n_k \to X$ sending the boundary to the constant at $x_0$?
- If not, is there a better definition of subdivided cubes for which the answer to (1) becomes "yes"?
It's nice that with the above definition, $D^n_{k+1}$ can be obtained by gluing together a bunch of copies of $D^n_k$ in an easy way. But perhaps this is too good to be true.
A trivial observation is that for $n=1$ the above cubical subdivision is basically the same as barycentric subdivision and the answer to (1) comes out as "yes".
